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Yes.
That would be the set of all non-zero numbers. If by number you actually meant whole numbers, that is integers, then it would be the set of all non-zero integers. They are also called Additive Inverses. For example, -5 is the additive inverse of 5, because 5 + (-5) = 0. Similarly, 7 is the additive inverse of -7 because (-7) + 7 = 0.
Every positive rational number and its negative are the two square roots of the same positive rational number.
Every rational number does.
No. Every real number is not a natural number. Real numbers are a collection of rational and irrational numbers.